be a positive integer. We start with 
and 
be the incircle and circumcircle of the acute triangle 
,
so that all of its sides are tangent to 
,
are both on 
.
and 
,
and 
,
and 
intersects on 
be integers greater than 
,
be positive integers not greater than 
.
not greater than ![\[ \gcd(a_1 + b_1, a_2 + b_2, \dots, a_m + b_m) < n, \]](http://latex.artofproblemsolving.com/7/4/1/7414caaacc9fd761961279f8a37ab6cd6af6b49a.png)
where 
denotes the greatest common divisor of 
.
be a fixed integer. Let 
and for all 
let 
.
for which sequence 
will be bounded?
denotes Euler’s Phi Function, meaning that 
gives the number of integers within the set 
that are relative primes to 
bounded if there exist a constant 
such that 
for all positive integers 
Y by
In an acute triangle 
with circumcircle 
and circumcenter 
, the circle 
is drawn, passing through the points 
, and 
together with its diameter 
, then the points 
and 
are chosen on the lines 
and 
, respectively, in such a way that the quadrilateral 
is a parallelogram.
Prove that the point of intersection of the lines
and 
lies on the circle .
with circumcircle 
and circumcenter 
, the circle 
is drawn, passing through the points 
, and 
together with its diameter 
, then the points 
and 
are chosen on the lines 
and 
, respectively, in such a way that the quadrilateral 
is a parallelogram.Prove that the point of intersection of the lines
and 
lies on the circle .
.
Therefore, lines 
and 
are tangent to 
and 
,
.
,
Since 
is a parallelogram, then
As a consequence,
and so
Since line 
is tangent to 
as points belonging to the sides of the triangle 
.
.
are collinear.
Also,
From 
and 
we get that 
are collinear.
Therefore, 
,
